On Spaces of Lipschitz Maps with Values in a Uniform Algebra (Researches on isometries as preserver problems and related topics)

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(1)52. On Spaces of Lipschitz Maps. with Values in a Uniform Algebra Kyohei Tanabe *. Master’s Program in Fundamental Sciences Graduate School of Science and Technology Niigata University In this paper, linear always means complex linear, especially Banach algebra always means complex Banach algebra. Isometries with respect to the s ‐norm between vector. valued Lipschitz spaces were studied by Hatori and Oi [2]. We prove a version of their results (Main Theorem A). There are literatures which study isometries with respect. to the. \max. ‐norm between vector valued Lipschitz spaces [1, 4]. In this paper, we. exhibit the form of isometries with respect to the. \max. ‐norm under an additional. condition (Main Theorem B) (cf. [7].).. 1. Definitions In this section, we introduce some basic definitions.. Definition 1.1. Let f :. Xarrow E. X. be a compact metric space and. E. a normed space. A map. is called a Lipschitz map if. (L(f):=) \sup_{x,y\in X,x\neq y}\frac{\Vert f(x)-f(y)\Vert_{E} {d(x,y)}<\infty. The number L(f) is called the Lipschitz constant of f . We shall denote by Lip (X, E) the space of all Lipschitz maps from *. X. into. E.. We write the space Lip (X, \mathbb{C}) just by. This work was supported by the Research Institute for Mathematical Sciences, a Joint Us‐. age/Research Center located in Kyoto University..

(2) 53 Lip (X) for simplification. There are several norms on the Lipschitz space Lip (X, E) : s. ‐norm \Vert. \Vert_{s} is defined by \Vert. \Vert_{rnax}=\max\{\Vert . \Vert {}_{\infty}L (.)\} . If. \Vert_{s}=\Vert B. \Vert_{\infty}+L(\cdot) , and max‐norm \Vert. is a Banach algebra, the space (Lip (X,. a Banach algebra, and the space (Lip (X,. B ),. \Vert_{7nax} by \Vert B ),. \Vert . \Vert_{s} ) is. \Vert . \Vert_{\max} ) is a Banach space (In general,. submultiplicativity needs not hold.).. 2. Main Theorem A The next is the theorem of isometries with respect to the s ‐norm between Lipschitz. spaces. In this section, we give an outline of the proof of this theorem.. Theorem 2.1 (Main Theorem A). For j=1,2 , let X_{j} be a compact metric space, Y_{j}. a compact Hausdorff space, and A_{j} a uniform algebra. If. U. : (Lip (X_{1}, A_{1}), \Vert\cdot\Vert_{s} ). arrow. (Lip (X_{2}, A_{2}), \Vert . \Vert_{s} ) is a unital surjective linear isometry, then there exist \bullet. a continuous map \psi : X_{2}\cross Ch (A_{2})arrow X_{1} such that for every y'\in Ch (A_{2}). \psi(\cdot, y') : X_{2}arrow X_{1} is a surjective isometry, and. \bullet. a homeomorphism. \tau. : Ch (A_{2})arrow Ch(A_{1}). such that (U(F)(x'))(y')=(F(\psi(x', y')))(\tau(y')) for every x'\in X_{2}, y'\in Ch (A_{2}) , and F\in Lip (X_{2}, A_{2}) .. Remark 2.2. A map. U. being unital means U(1)=1 . The space Ch (A) denotes. a Choquet boundary of A. (If. Y. is a compact metric space and. A. is a subspace of. (C(Y), \Vert\cdot\Vert_{\infty}) , Ch (A)= { y\in Y|\tau_{y}\in ext \{\varphi\in A^{*}|\Vert\varphi\Vert=\varphi(1)=1\} } where. \tau_{y}. is. the evaluation map at y\in Y. ) Outline of the proof of the Main Theorem A. First, we regard Lip (X_{j}, A_{j}) as a subspace of C(X_{j}\cross Y_{j}) . We apply a theorem of. Jarosz [3], then we find that. U. is an isometry also with respect to the supremum norm.. Using partition of unity, we find that the uniform closure of Lip (X_{j}, A_{j}) coincides with. C(X_{j}, A_{j}) .. So we can extend U from. C(X_{1}, A_{1}). onto. C(X_{2}, A_{2}). which is a. unital surjective linear isometry with respect to the supremum norm. We denote this.

(3) 54 map by \tilde{U}^{\infty} We define maps S. by S(\varphi'). :=\varphi'\circ\tilde{U}^{\infty}. \{\varphi'\in C(X_{2}, A_{2})^{*}|\Vert\varphi'\Vert=\varphi'(1)=1\} arrow\{\varphi\in C(X_{1}, A_{1})^{*}|\Vert\varphi\Vert=\varphi(1)=1\} :. and. \{\varphi\in C(X_{1}, A_{1})^{*}|\Vert\varphi\Vert=\varphi(1)=1\} arrow\{\varphi'\in C(X_{2}, A_{2})^{*}|\Vert\varphi'\Vert=\varphi'(1)=1\} S'. by S'(\varphi) S,. :. :=\varphi o(\tilde{U}^{\infty})^{-1}. Then,. S. and S' are well‐defined, S' is an inverse map of. and S is a w^{*} ‐homeomorphism.. For j=1,2 , we define a set. K_{j}. :=. ext. \{\varphi\in C(X_{j}, A_{j})^{*}|\Vert\varphi\Vert=\varphi(1)=1\}.. Then we find that S(K_{2})=K_{1} by some easy argument of extreme points. We note that the Choquet boundary of C(X_{j}, A_{j}) coincides with X_{j}\cross Ch(A_{j}) . If we define a homeomorphism \Phi_{j} : X_{j}\cross Ch (A_{j})arrow K_{j} by the evaluation at (x, y) for j=1,2 , then the map. \Phi_{j}(x, y)=\varphi_{(x,y)} where. \Phi_{1}^{-1}oSo\Phi_{2}. \varphi_{(x,y)} is. is a homeomorphism. between X_{2}\cross Ch (A_{2}) and X_{1}\cross Ch (A_{1}) . So we can define continuous maps \psi_{1} :. X_{2}\cross Ch(A_{2})arrow X_{1}, \psi_{2} : X_{2}\cross Ch(A_{2})arrow Ch(A_{1}) by (\psi_{1}, \psi_{2})=\Phi_{1}^{-1}oSo\Phi_{2} . By the similar way, we consider the homeomorphism. \Phi_{2}^{-1}oS^{-1}o\Phi_{1}. between X_{1}\cross Ch(A_{1}). and X_{2}\cross Ch (A_{2}) , and define continuous maps \psi í : X_{1}\cross Ch (A_{1})arrow X_{2}, \psi_{2}' : X_{1}\cross Ch(A_{1})arrow Ch(A_{2}) by ( \psi í, \psi_{2}' ) =\Phi_{2}^{-1}oS^{-1}o\Phi_{1} . Then for every x'\in X_{2} and y'\in Ch(A_{2}),. ((U(F))(x'))(y')=S(\varphi_{(x,y)}')(F)=(F(\psi_{1}(x', y')))(\psi_{2}(x', y')). . We. shall observe the maps \psi_{1}, \psi_{2}.. At the first, We show that the map \psi_{2} needs not depend on the first variable x'\in X_{2} , that is, the equality \psi_{2} (xí, y' ) =\psi_{2}(x_{2}', y') holds for any xí, x_{2}'\in X_{2} and y'\in Ch (A_{2}) . Suppose that there are x_{1}^{\circ}\neq X_{2}^{\circ}\in X_{2} and y^{\circ}\in Ch (A_{2}) such that. \psi_{2}(x_{1}^{\circ}, y^{\circ})\neq\psi_{2}(x_{2}^{\circ}, y^{\circ}) .. Then there is. h\in A_{1} such that h(\psi_{2}(x_{1}^{\circ}, y^{\circ}))\neq. h(\psi_{2}(x_{2}^{\circ}, y^{\circ})) since A_{1} is a uniform algebra. By the direct computation, we assert that L(1\otimes h)=0 and L(U(1\otimes h))\neq 0 . On the other hand. U. preserves the Lip‐. schitz constant because U preserves the s ‐norm and the supremum norm. This is a. contradiction. Hence the map \psi_{2} needs not depend on the first variable.. Then we define continuous maps. \tau. (y'\in Ch(A_{2})) for some x'\in X_{2} , and. : Ch (A_{2})arrow Ch (A_{1}) by \tau(y')=\psi_{2}(x', y') \tau'. : Ch (A_{1})arrow Ch (A_{2}) by \tau'(y)=\psi_{2}'(x, y).

(4) 55 (y\in Ch(A_{1})) for some x\in X_{1} . We can check that the map. \tau. is a homeomorphism. between Ch (A_{2}) and Ch (A_{1}) . Moreover, \tau' is an inverse map of. \tau. . On the other. hand, the maps \psi_{1}(\cdot, y') : X_{2}arrow X_{1} and \psi í : X_{1}arrow X_{2} are bijective for each y'\in Ch (A_{2}) and \psi í. (\cdot, \tau^{-1}(y))^{-1}. y\in. Ch (A_{1}) respectively. Moreover, \psi_{1}(\cdot, y') =\psi í. (\cdot, \tau(y'))^{-1} and. hold for each y'\in Ch(A_{2}) and y\in Ch(A_{1}) respectively.. These indicate that it is sufficient to show that \psi_{1} ( , yÓ) : X_{2}arrow X_{1} is a contractive \cdot. map for each y\'{O}\in Ch(A_{2}) which proves the Main Theorem A. Take y\'{O}\in Ch(A_{2}) arbi‐. trarily. We prove that. d. ( \psi_{1} (xí, yÓ), \psi_{1}(x_{2}', y\'{O}) ) \leq d (xí, x_{2}' ) for every distinct xí, x_{2}'\in. . We define a function f_{\psi_{1}()}x_{2},y_{0} : X_{1}arrow \mathbb{C} by f_{\psi_{1}(x_{2,} ,yÓ) (x)=d(x, \psi_{1} ( x_{2}' , yÓ) ) for x\in X_{1} . Then f_{\psi_{1}()}x_{2},y_{0} is in Lip (X_{1}) and L(f_{\psi_{1}()}x_{2},y_{0})=1 . Therefore, X_{2}. d. =d. ( \psi_{1} (xí, yÓ) \psi_{1}(x_{2}' , yÓ)) ( \psi_{1} (xí, yÓ) , \psi_{1}(x_{2}', y\'{O}) ). -d. ( \psi_{1}(x_{2}' , yÓ) \psi_{1}(x_{2}' , yÓ)). =|f_{\psi_{1}(x_{2}'} ,yÓ ) (\psi_{1} (xí, y\'{O}))-f_{\psi_{1}(x_{2}'} ,yÓ ) ( \psi_{1}(x_{2}' , yÓ)) | =| ( ( f_{\psi_{1}(x_{2}'} ,yó) \otimes 1) ( \psi_{1} (xí, yÓ)) ) ( (yÓ)) — ( f_{\psi_{1}(x_{2}'} ,yÓ) \otimes 1) ( \psi_{1}(x_{2}' , yÓ)) ) ( (yÓ)) | =| ((U(f_{\psi_{1}()}x_{2}',y_{0}'\otimes 1)) (xí) ) (y\'{O})-((U(f_{\psi_{1}(x_{2}'} ,yÓ) \otimes 1))(x_{2}')) (yÓ) | \leq\Vert (U (f_{\psi_{1}(x_{2}'},y\'{O})\otimes 1)) (x\'{i})-(U(f_{\psi_{1}(x_{2}'} , yÓ) \otimes 1))(x_{2}')\Vert_{\infty(Y_{2})} \leq d (xí, x_{2}' ) L(U ( f_{\psi_{1}(x_{2} , ,yÓ) \otimes 1)) . \tau. Since. U. \tau. preserves the Lipschitz constant, we have. L(U(f_{\psi_{1}()}x_{2}',y_{0}'\otimes 1))=L(f_{\psi_{1}()}x_{2}',y_{0}'\otimes 1)=L(f_{\psi_{1}()}x_{2}',y_{0}')=1. Hence we have d. ( \psi_{1} (xí, yÓ),. \psi_{1}(x_{2}', y\'{O}) ). \leq d. (xí, x_{2}' ). =d. (xí, x_{2}' ). L(U(f_{\psi_{1}()}x_{2}',y_{0}'\otimes 1)). and \psi_{1} ( , yÓ) is a contractive map. We complete the outline of the proof of the Main \cdot. Theorem A. \square.

(5) 56. 3. Main Theorem B In this section, we consider isometries with respect to the. \max. ‐norm between Lips‐. chitz spaces. We exhibit the Main Theorem B and give an outline of the proof of this theorem.. Definition 3.1 ( K pair). Let X_{1} and X_{2} be compact metric spaces. We say that the ordered pair (X_{1}, X_{2}) of these two sets is a. K. pair if the following two conditions are. satisfied.. (K1) For j=1,2 , if we take any. \bullet. X_{j}. such that. d(x_{1}, x_{\mathring{1}})<1,. x_{1},. x_{2}\in X_{j} , there are finitely many x_{1}^{\circ},. d(x_{i}^{\circ}, x_{i+1}^{\circ})<1(i=1, \ldots n-1), d(x_{\mathring{n}}, x_{2})<1.. (K2) For any bijection \psi : X_{2}arrow X_{1} and positive. \bullet. ment holds; if xí, x_{2}'\in X_{2} and d. x_{n}^{\circ}\in. d. (xí, x_{2}' ). <\varepsilon. \varepsilon. , the following state‐. implies that d(\psi (xí), \psi(x_{2}'))=. (xí, x_{2}' ), then \psi is an isometry.. Theorem 3.2 (Main Theorem B). For j=1,2 , let X_{j} be a compact metric space, Y_{j} a compact Hausdorff space, and A_{j} a uniform algebra. We assume that (X_{1}, X_{2}). is a. K. pair. If. U. : (Lip (X_{1}, A_{1}), \Vert . \Vert_{\max} ). arrow. (Lip (X_{2}, A_{2}), \Vert . \Vert_{\max} ) is a unital. surjective linear isometry, then there exist \bullet. a continuous map \psi : X_{2}\cross Ch (A_{2})arrow X_{1} such that for every y'\in Ch (A_{2}). \psi(\cdot, y') : X_{2}arrow X_{1} is a surjective isometry, and. \bullet. a homeomorphism. \tau. : Ch (A_{2})arrow Ch(A_{1}). such that (U(F)(x'))(y')=(F(\psi(x', y')))(\tau(y')) for every x'\in X_{2}, y'\in Ch (A_{2}) , and F\in Lip (X_{2}, A_{2}) . Outline of the proof of the Main Theorem B. We can prove by the same way as the outline of the proof of Theorem 2.1 that there are continuous maps \psi_{1} : X_{2}\cross Ch(A_{2})arrow X_{1}, \psi_{2} : X_{2}\cross Ch(A_{2})arrow Ch(A_{1}) such. that for every x'\in X_{2}, y'\in Ch(A_{2}), ((U(F))(x'))(y')=(F(\psi_{1}(x', y')))(\psi_{2}(x', y')) holds..

(6) 57 At the first, we show that the map \psi_{2} needs not depend on the first variable x'\in X_{2} , that is, the equality \psi_{2} (xí, y' ) =\psi_{2}(x_{2}', y') holds for any xí, x_{2}'\in X_{2}. and y'\in Ch (A_{2}) .. By the condition (K1) , it suffices to show that the equality. \psi_{2} (xí, y' ) =\psi_{2}(x_{2}', y') holds for every xí, x_{2}'\in X_{2} with. d. (xí, x_{2}' ). <1. and y'\in. Ch (A_{2}) . If not, there exist x_{1}^{\circ}, x_{2}^{\circ}\in X_{2} with d(x_{1}^{\circ}, x_{2}^{\circ})<1 and y^{\circ}\in Ch (A_{2}) such that \psi_{2}(x_{1}^{\circ}, y^{\circ})\neq\psi_{2}(x_{2}^{\circ}, y^{\circ}) . Let V\subset Y_{1} of. \varepsilon_{0}=\frac{1-d(x_{1}^{\circ},x_{2}^{\circ})}{2} .. We take an open neighborhood. \psi_{2}(x_{1}^{\circ}, y^{\circ}) which doesn’t contain \psi_{2}(x_{2}^{\circ}, y^{\circ}) . Then there is a peaking. function h\in A_{1} such that h(\psi_{2}(x_{1}^{\circ}, y^{\circ}))=1 , and |h(y)|<\varepsilon_{0} for every y\in Y_{1}\backslash V. Especially |h(\psi_{2}(x_{2}^{\circ}, y^{\circ}))|<\varepsilon_{0} . It is clear that \Vert 1\otimes h\Vert_{7nax}=1 . On the other hand,. L(U(1 \otimes h) \geq\frac{|U(1\otimes h)(x_{1}^{\circ},y^{\circ})-U(1\otimes h)(x_{2}^{\circ},y^{\circ})|}{d(x_{1}^{\circ},x_{2}^{\circ})} = \frac{|h(\psi_{2}(x_{1}^{\circ},y^{\circ}) -h(\psi_{2}(x_{2}^{\circ}, y^{\circ}) |}{d(x_{1}^{\circ},x_{2}^{\circ}) >\frac{1-2\varepsilon_{0} {d(x_{\mathring{1} ,x_{2}^{\circ}) =1 holds, hence we get \Vert U(1\otimes h)\Vert_{7nax}>1 . This contradicts to the fact that the. \max. U. preserves. ‐norm. Thus \psi_{2} needs not depend on the first variable.. By the Theorem of Jarosz [3], norm. We can extend. U. U. is also an isometry with respect to the supremum. from the uniform closure of Lip (X_{1}, A_{1}) , which is C(X_{1}, A_{1}) ,. onto the uniform closure of Lip (X_{2}, A_{2}) , which is C(X_{2}, A_{2}) , that is a unital surjective linear isometry with respect to the supremum norm. Since C(X_{j}, A_{j}) is a uniform. algebra, a theorem of Nagasawa [5] yields that. U. is multiplicative. For each y'\in. Ch (A_{2}) , we define a map U_{y'} : Lip (X_{1})arrow Lip(X_{2}) by. U_{y'}(f)=((U(f\otimes 1))(\cdot))(y') for f\in Lip (X_{1}) . U_{y'} is a unital homomorphism. So by [6, Theorem 5‐1] , there is a Lipschitz map \phi_{y'} : X_{2}arrow X_{1} such that U_{y'}(f)=f\circ\phi_{y'} for every f\in Lip (X_{1}) . It is easy to check the equality \phi_{y'}=\psi_{1}(\cdot, y') . Hence \psi_{1}(\cdot, y') is a Lipschitz map.. Next we prove that \psi_{1}(\cdot, y') : X_{2}. y'\in Ch (A_{2}) .. arrow X_{1} is a surjective isometry for each. \varepsilon_{0}=\frac{1}{\max\{1,L(\psi_{1}(\cdot,y) \} .. Let. By (K2) and the descriptions in the. outline of the proof of Theorem 2.1, it suffices to show that for every xí, x_{2}'\in X_{2} with. d. (xí, x_{2}' ). We define. <. \varepsilon_{0}. f_{\psi_{1}()}x_{2},’ y. , the equality \in. Lip (X_{1}) by. d. (xí, x_{2}' ). \geq. d(\psi_{1} (xí, y' ), \psi_{1}(x_{2}', y')) holds.. f_{\psi_{1}()}x_{2},y'(x)= \min\{d(x, \psi_{1}(x_{2}', y')), 1\}. for.

(7) 58. L(f_{\psi_{1}()}x_{2},y'). x\in X_{1} , then we have. tion of d. \varepsilon_{0}. , we get. d. \leq 1,. \Vert f_{\psi_{1}()}x_{2},y'\Vert_{\infty}\leq 1 .. (\psi_{1} (xí, y' ), \psi_{1}(x_{2}', y'))\leq 1 .. Hence. By the defini‐. f_{\psi_{1}()}x_{2},y' ( \psi_{1} (xí,. (\psi_{1} (xí, y' ), \psi_{1}(x_{2}', y')) , and. y') ). =. d(\psi_{1} (xí, y'), \psi_{1}(x_{2}', y')). =|f_{\psi_{1}(x_{2}',y^{\prime)} ( \psi_{1} (xí, y') ) -f_{\psi_{1}(x_{2}',y^{\prime)} (\psi_{1}(x_{2}', y') | =| (U(f_{\psi_{1}()}x_{2}',y'\otimes 1) (xí) ) (y')-(U(f_{\psi_{1}()}x_{2}',y'\otimes 1)(x_{2}'))(y')| \leq\Vert U(f_{\psi_{1}()}x_{2}',y'\otimes 1) ( í)— (f_{\psi_{1}(x_{2}',y^{\prime)} \otimes 1)(x_{2}')\Vert_{\infty(Y_{2})} \leq L(U(f_{\psi_{1}()}x_{2},y'\otimes 1))d (xí, x_{2}' ) \leq d (xí, x_{2}' ). x. U. Thus we get the desired inequality. Now we complete the outline of the proof of the Main Theorem B. \square. In the next section, we observe some examples of without the condition,. 4. K. K. K. pairs, and Main Theorem B. pair.. pairs. In the Main Theorem B, we assume that (X_{1}, X_{2}) is a examples of. K. K. pair.. We give some. pairs.. Example 4.1.. 1. If a<b , the pair of closed intervals ([a, b], [a, b]) is a. K. pair.. 2. Let. \overline{\mathbb{D}}=\{z\in \mathbb{C}||z|\leq 1\} with the usual distance, then (\overline{\mathb {D} ,\overline{\mathb {D} ) is a. 3. Let. K=( \{0\}\cross[-1,1])\cup(\{(t, \frac{1}{2}t)|0\leq t\leq 2\})\cup(\{(t, - \frac{1}{2}t)|0\leq t\leq 2\})\subset \mathbb{R}^{2}. with the usual distance, then (K, K) is a. K. K. pairs.. Example 4.2. Let X_{1}=X_{2}=Y_{1}=Y_{2}=\{a, b\} where the distance of. 2, then (X_{1}, X_{2}) is not a. pair.. pair.. It is not difficult to check that these three pairs above are. K. K. a. and. b. is. pair because it doesn’t satisfy (K1) . We define a map. \phi:X_{2}\cross Y_{2}arrow X_{1}\cross Y_{1} by. \phi((a, a))=(a, a), \phi((a, b))=(b, a).

(8) 59 \phi((b, a))=(a, b), \phi((b, b))=(b, b) and maps \psi_{1} : X_{2}\cross Y_{2}arrow X_{1}, \psi_{2} : X_{2}\cross Y_{2}arrow Y_{1} by \phi=(\psi_{1}, \psi_{2}) . Let. U. :. Lip (X_{1}, C(Y_{1}))arrow Lip(X_{2}, C(Y_{2})) be. ((U(F))(x'))(y')=(F(\psi_{1}(x', y')))(\psi_{2}(x', y')) for x'\in X_{2}, y'\in Y_{2} , and. F\in. isometry with respect to the. Lip (X_{1}, C(Y_{1})) . Then. \max. U. is a unital surjective linear. ‐norm. Actually for every. F\in. Lip (X_{j}, C(Y_{j})) ,. L(F)= \frac{\Vert F(a)-F(b)\Vert_{\infty} {2}\leq\Vert F\Vert_{\infty}. Hence the. \max. ‐norm coincides with the supremum norm in this case. The map. clearly an isometry with respect to the supremum norm. But. U. U. is. cannot be represented. as the form in Theorem 3.2.. Example 4.3. Let. H=(\{0\}\cross[-1,1])\cup([0,3]\cross\{0\})\cup(\{3\}\cross[-1,1])\subset \mathbb{R}^{2} with. the usual distance. Then (H, H) is not a. K. pair. To prove this, we define a bijection. \psi:Harrow H by. \psi(x,y)=\{ begin{ar y}{l (x,y) (x,y)\in\{0\} cros [-1, ]) (x,y) (x,y)\in[0,3]\cros \{0\}) (x,-y) (x,y)\in\{3\} cros [-1, ]). \end{ar y}. Then d((x_{1}, y_{1}), (x_{2}, y_{2}))<2 implies that d((x_{1}, y_{1}), (x_{2}, y_{2}))=d(\psi((x_{1}, y_{1})), \psi((x_{2}, y_{2}))). but \psi is not an isometry. Let U. Y. be any compact Hausdorff space. We define a map. : Lip (H, C(Y))arrow Lip(H, C(Y)) by. ((U(F))(x'))(y')=(F(\psi(x')))(y') for x'\in H, y'\in Y , and form in Theorem 3.2, but. F\in U. Lip (H, C(Y)) .. This. U. is not represented by the. is a unital surjective linear isometry with respect to the. \max‐norm.. References [1] Botelho, Fernanda; Fleming, Richard J.; Jamison, James E. Extreme points and isometries on vector‐valued Lipschitz spaces. J. Math. Anal. Appl. 381 (2011), no. 2, 821‐832..

(9) 60 [2] Hatori, Osamu; Oi, Shiho, Hermitian operators on Banach algebras of vector‐ valued Lipschitz maps. J. Math. Anal. Appl. 452 (2017), no. 1, 378‐387.. [3] Jarosz, Krzysztof, Isometries in semisimple, commutative Banach algebras. Proc. Amer. Math. Soc. 94 (1985), no. 1, 65‐71.. [4] Jiménez‐Vargas, A.; Villegas‐Vallecillos, Moisés, Linear isometries between spaces of vector‐valued Lipschitz functions. Proc. Amer. Math. Soc. 137 (2009), no. 4, 1381‐1388.. [5] Nagasawa, Masao, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions. Ko‐dai Math. Sem. Rep. 11 (1959), 182‐ 188. 46.00. [6] Sherbert, Donald R. Banach algebras of Lipschitz functions. Pacific J. Math. 13 (1963), 1387‐1399. [7] Tanabe, Kyohei, Isometries on Spaces of Lipschitz Maps with Values in a Uniform Algebra, master’s thesis in Niigata University (2019).

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